Is it possible for an object to follow multiple Clamp To constraints?
Add a Follow path constraint to the Cylinder, with the bezierCircle object as the target. Enable Follow Curve and set forward to -Z and Up as Y.
(For this to work the curve and the cylinder must have their origins on the same location)

The cylinder follows the path now, as demonstrated here changing the offset values:



Align your Cube.001 sliding object to the cylinder and add a child of constraint, with the cylinder as target. Disable all the boxes except for Location on the Z axis.



To control the slider change the offset values on the follow path constraint for the cylinder.